Much has been written about the process of selecting motors for applications in motion control. Motor selection can be complicated. The decisions to be made include whether to employ brushless or brushed dc, or stepper motors. Stepper motors are generally considered specialpurpose devices because they can’t handle large inertial loads very well. And operation near their maximum torque ratings can take some finesse.

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In contrast, dc motors are usually considered more general-purpose motioncontrol devices. They are a frequent choice for supplying variable speeds when precise speed regulation is not critical. They can handle brief overloads as well as frequent stops and starts. Dc motors are usually rated at a single speed and torque, and they can operate at rated conditions continuously. There are basic calculations that can point the way to using the right dc motor in the right situation.

Parameters that define the best motor type for an application include the mechanical output power, the shaft bearing system, the commutation system, and the possible combinations with gearheads and sensors. The most important criteria include the required speed and torque and the commutation system.

Of course, equations define these relationships. In this article, variables on the motor shaft (output) are identified with the subscripts Mot. For example, n_{Mot} gives the required motor speed. Parameters that describe the qualities of the motor have no special additional subscript. For example, *n _{o} *stands for the motor’s no-load speed.

We’ll discuss speed and torque requirements first. The maximum speed on the motor shaft n_{maxMot} should be below the maximum permissible speed of the motor,*n _{max} or *

*n _{maxMot} < n_{max}*

As a rule, expected useful life drops as motor speed rises because of the greater load on the bearings. The motor generates more noise as well. In brush-dc motors, there is a higher mechanical wear of the brush system. These effects are particularly pronounced at speeds above the maximum permissible speed. The effective torque, *M _{rmsMot}* that the application demands must be less than the rated torque of the motor,

*M*

_{N}or *M _{rmsMot} < M_{N}*

Rated torque is derived from the rated current, *I _{N}* which is selected to equal the maximum continuous current of the motor (from a thermal point of view). Note some motor manufacturers include motor friction losses with the permissible continuous torques. (Maxon is one manufacturer that follows this policy. The friction losses of Maxon motors are quite low.) The friction losses of graphite brushes and the iron losses of brushless electronically commutated (EC) motors depend on speed. The result is the boundary between continuous and temporary operating ranges tends to curve somewhat such that higher rotational speeds demand slightly lower current. This is simply because there is less frictional loss at the higher speeds.

Of course, the motor must be able to produce enough torque for the application. This means during startup, torque demands should not exceed the motor’s stall torque at the rated voltage, *M _{H}*. However, this is not exactly true on larger motors with a correspondingly higher stall torque. These motors can typically handle brief peak torques up to about four times the rated torque,

*M*without any problems:

_{N} *M _{maxMot} < 4 • M_{N} or M_{MaxMot} < M_{H}*

A more detailed analysis would take into account the extent and duration of the overload, as well as the winding and ambient temperatures. In highly dynamic applications, the additional torque needed for motor acceleration must be included in the calculation of the operating points:

*M _{Motα} • (π/30) • (Δn_{Mot}/Δt_{α })*

where *Δt _{α}* = time duration and

*J*= rotor moment of inertia. The effective load must then be recalculated.

_{rot} Some motor selection considerations involve the combined mass inertia of motor and load. Motors having a low mass inertia can typically handle high reflected-load mass inertias. In the case of Maxon motors, for example, reflected-load mass inertias up to 10 times the rotor inertia, *J _{rot}* are acceptable, and even higher levels are okay in some instances. The ratio should be low for highly dynamic drives, as illustrated by the relationship between the mechanical time constant of the motor and the reflected load-moment of inertia,

*J*.

^{*}_{L} *Τ _{mech} = Τ_{ }m • (1+ J^{*}_{L} / J _{rot} )*

where *Τ _{m} *= the mechanical time constant of the motor without load.

A load inertia of the same magnitude as *J _{rot} *doubles the reaction time of the motor.

When the motor load has a large inertia, the load/ motor coupling method becomes important. Motor shafts and couplings act like torsional springs and must be appropriately stiff so the load doesn’t begin to swing in highly dynamic systems. A typical means of boosting stiffness it to specify a thicker or shorter motor shaft. This, in turn, could force the use of a larger motor.

Systems that must accelerate a large load inertia *J _{L} *rapidly may incorporate gearing to do so more easily. The selection of a suitable drive reduction ratio can minimize the needed motor torque. Here, the optimum reduction ratio I results from the ratio of the mass inertias of the load and the motor

for linear motion and reduction ratio.

**Factoring in torque**

There are a number of factors that can influence a motor’s rated torque. Motor manufacturers usually specify rated torque under conditions that are well spelled out. Take, for example, the rated torque in a Maxon motor. *M _{N} *represents the maximum permissible continuous torque. The rated torque varies as a function of the ambient temperature,

*T*, and mounting conditions; a higher ambient temperature results in less-efficient dissipation of heat and consequently a lower continuous torque.

_{Amb} Similarly, the details of thermal coupling influence the thermal resistance, *R _{th2}*, between housing and ambient and, thus, the rated torque. Better heat dissipation lowers

*R*and permits a higher rated torque. The thermal resistance,

_{th2}*R*, can easily be cut in half by forced cooling or thermal coupling to a heat-conducting (metal) heat sink.

_{th2}The selection of a commutation system is important. The primary decision is between mechanical commutation (dc motor with brushes) and electronic commutation (brushless dc motor, BLDC). Considerations of life expectancy, reliability, simplicity of actuation, and maximum speed play a role in this process. Special environmental conditions such as operation in a vacuum or the ability to withstand sterilization cab also be a consideration.

**Getting the windings right**

Before one can specify the appropriate winding for a motor type, it is necessary to understand the winding electrical qualities and how they influence motor performance. Most fundamentally, windings affect the speed-versus-torque behavior. The speed-torque line describes the possible operating states (working points) of a motor at a given applied motor voltage, *U _{Mot}*. On dc and EC motors, this characteristic line is represented in the standard speed versus torque diagram as a linear relationship that runs from the no-load speed,

*n*(torque = 0), to the stall torque,

_{0}*M*(speed = 0). Performance follows the equation:

_{H} *n = k _{n} • U_{Mot} – (Δn/ΔM) • M*

where * nΔ/ΔM* = the slope of the speed-torque line and *k _{n} *= the speed constant of the motor.

The speed-torque line describes how the motor progressively slows as the load rises, until it ultimately halts at the stall torque. From a torque point of view the motor exhibits the maximum torque at stall, and the faster it turns the less torque it produces.

We first calculate the needed no-load speed. Approximate the average slope of the speed-torque line of the A-max 12 motor type to 8,100 rpm/mNm (line 14 of the table, below.) = 14,000 + 8,100 • 0.25 The desired speed constant of the motor is then = 16,000/12 A comparison with the possible speed constants (line 13 above) shows the motor best suited for this requirement ( = 0.25/6.92 + 0.00587 |

In motor catalogs, voltage-dependent values are often given at the rated voltage *U _{N}* (understood as a reference voltage), which is not necessarily the voltage at which the motor must operate. The influence of the applied motor voltage,

*UMot*, on the speed-torque line is a parallel shift: upwards at higher voltages and downwards at lower voltages. Slope remains the same. The no-load speed and stall torque shift accordingly. The no-load speed is easily calculated using the applied motor voltage and the speed constant:

*n _{0} = k_{n} • U_{Mot}*

Motor losses are considered negligible for this first approximation. Note this simple relationship between motor speed and applied voltage is valid only at no-load operation.

The torque generated by the motor is proportional to the current through the windings. So one can draw a set of characteristic lines representing speed-versus-torque for the various possible windings of a motor series, with each winding set at its specific rated voltage. For these curves, rated voltages are chosen to give a no-load speed that is similar for all the different windings. Plotting these characteristic lines at the same motor voltage yields a group of parallel straight lines. Thus all the lines have the same slope or gradient, even at different speed constants (*k _{n}*).

The slope or gradient of the speed torque line is a measure of motor performance. A flat speedtorque line with a gradual slope corresponds to a powerful motor where speed varies only slightly as the load increases. Less-powerful motors have a steeper speed-torque line with a correspondingly higher slope.

The slope of the speed-torque line is a constant determined solely by design variables: the geometric arrangement and dimensions of the winding and magnetic circuit and the resulting magnetic flux density in the air gap. The number of windings and the resistance of the winding also are part of the calculation. However, their influence cancels over a series of windings for most commercial motors. So the slope of the speed-torque line may be considered to represent a mechanical constant. Motors with the same speed-torque slope will exhibit the same speed response under load.

Larger motors frequently use the constant k with units of *Nm/W ^{1/2} *as a comparative measurement for motor performance. A higher k value indicates a more-powerful motor. The slope of the characteristic line has the same meaning as k, and they are related by

*Δn/ΔM α 1/k ^{2}*

although Δn/ΔM is more practical for the calculations.

For example, a speed-torque line slope of 5 rpm/mNm means that at constant applied voltage a rise in the required torque by 10 mNm results in a speed drop by 10 mNm • 5 rpm/ mNm = 50 rpm.

The slope of the speed-torque line of a motor can be calculated from the no-load speed, n_{0}, and the stall torque, *M _{H}*, values listed in the motor data sheets. The following equation applies:

*Δn/ΔM = n _{0}/M_{H}*

If the motor data sheet is unavailable, the slope of the speed-torque line can be calculated from a measurement of the no-load speed, n0, at the motor voltage, U, and from the motor resistance, *R*:

*Δn/ΔM = π/30 • [(R • n _{0}^{2})/U^{2}]*

The electrical resistance, *R*, of the motor can be determined by measuring motor current with the motor shaft locked. This current is starting current, *I _{A}*:

*R = U/I _{A}*

The objective of winding selection is to find a motor with the speed-torque line that covers all the working points dictated by the load at a given voltage, and especially at the extreme working point. The speed-torque line must extend above this point at the maximum available voltage. All other working points can then be reached by reducing the motor voltage. This extreme working point falls at the end of the acceleration process, where both the torque and speed are at maximum.

Factors that go into the selection of commutation and bearing systems include that these systems typically need a long life expectancy measured both in operating hours and number of duty cycles. Also typical is a requirement for cyclical motion in both directions of rotation. This sort of life often is available only with a brushless-EC motor and preloaded ball bearings – generally in combination with a feedback sensor. Another application that makes an interesting example is that of a billboard drive, where ads rotate and change at specified intervals. Typical operating requirements typically include around-the-clock start-stop operation with a cycle time of 10 sec (e.g., 2 sec on, 8 sec off). Over an operating life of 10 years, this comes to approximately 30 million duty cycles and a total on-time of 17,500 hr. Factors that influence the selection of commutation and bearing systems include the need for a long operating life and cyclical motion in both directions of rotation. Long life expectancy probably makes BLDC motors and preloaded ball bearings the first choice. The low speeds of the load make a ceramic gearhead a possibility. Moreover, the motor’s Hall sensors used for commutation might be sufficient for providing positioning feedback, given a gear reduction ratio that is sufficiently high. |

For a methodical determination of the appropriate windings, assume the optimum winding for an application has a speed-torque line that runs directly through the extreme working point (*M _{maxMot}*,

*n*) at the maximum possible motor voltage, U Mot . We next assume an average speed-torque line slope for the motor type. We can calculate a no-load speed,

_{maxMo}t*n*

_{0,theor}, as an auxiliary variable:

*n _{0,theor} = n_{maxMot} + (Δn/Δ M)_{avg} • M_{maxMot}*

Note that a more-conservative approach takes the slope of the least powerful winding of the motor type considered (i.e., the one with the steepest slope). Accordingly, n_{0,theor} becomes greater. The motor must hit this theoretical no- load speed with the maximum voltage, *U _{Mot}*. This is equivalent to saying that the speed constant, kn, of the selected motor must exceed the ratio of the no-load speed to the motor voltage:

*k _{n} > n_{0,theor}/U_{Mot}*

To this point, we have not considered tolerances in the load, drive, motor, controller and power supply. One way to factor in a tolerance is by selecting a winding with a speed constant that is higher by, say, 20%:

*k _{n} > 1.2 • [n_{0,theor}/U_{Mot} ] *

*= 1.2/U _{Mot} • [ n_{maxMot} + (Δn/ΔM)_{avg} • M_{maxMot} ]*

The final step is to verify there is enough current available for the motor winding to produce the required output torque for both intermittent and continuous operation. This information can be calculated from the torque constant *k _{M} *and the fact that motor current is proportional to the torque produced. Calculations of current must also factor in losses in the motor. These are expressed by the no-load current

*I0,*which depends slightly on speed. For estimations of sufficient current, however, this dependency can be ignored. Thus, in each time cycle the available current,

*I*, must exceed the current needed to produce the load torque (including the no-load current of the motor):

_{Mot} *I _{Mot} > ( M_{maxMot} /k_{M}) + I_{0}*

The current, *I _{Mot}*, available at the motor is set by the power supply and the controller.